For Fall/Spring tutorials, see this page.
Welcome Message
To enroll in a summer tutorial, please email Cindy Jimenez
(cindy@math.harvard.edu) by the evening of Monday, May 22 2019, giving her
an ordered preference list for the tutorials you wish to enroll in.
Note that you can enroll in more than one tutorial if space is available.
An overview of the tutorials is given on Wednesday, May 1, 4 PM in SC 507.
The summer tutorial program offers some interesting mathematics to
those of you who will be in the Boston area during July and August. The
tutorials will run for six weeks, meeting twice or three times per
week in the evenings (so as not to interfere with day time jobs). The
first tutorial (Topological Data Analysis) will
start early in July or late in June, and run to mid August. The precise
starting dates and meeting times will be arranged for the convenience of
the participants once the tutorial roster is set. The tutorial on Elliptic
Curves will start on July 15 and run till the last week in August.
The tutorial on knot invariants starts in early June and runs until mid or
late July.
The format of each tutorial will be much like that of the term-time
tutorials, with the tutorial leader lecturing in the first few meetings
and students presenting later on. Unlike the term-time tutorials, the
summer tutorials have no official Harvard status: you will not receive
either Harvard or concentration credit for them. Moreover, enrollment
in the tutorial does not qualify you for any Harvard-related perks
(such as a place to live). However, the Math Department will pay each
Harvard College student participant a stipend, and you can hand in your
final paper from the tutorial for your junior paper requirement for the
Math Concentration.
The topics and leaders of the tutorials this summer are:
Topological Data Analysis
Taught by
Jun-Hou Fung
Topological data analysis (TDA) is one of the hottest new developments
in visualizing and interpreting data. TDA brings together ideas from
category theory, algebraic topology, representation theory, metric
geometry, probability theory, and statistics to find and analyze
patterns in complex high-dimensional data sets. One key tool in this
subject is persistent homology, which is a way to study the shape of
data simultaneously at multiple feature scales. In this tutorial,
we will introduce the mathematics underpinning this theory from the
very basics to the frontiers of research. Topics include simplicial
complexes associated to data, persistence modules, structure theory
for barcodes, and stability results for persistent homology. If time
allows, we may also discuss further topics such as Morse theory, manifold
learning, and statistical inference in TDA, depending on the participants'
backgrounds and interests. In recent years, TDA flourished in biological
applications, especially with genomic data, and we will survey some of
these recent applications of TDA. While the tutorial will focus on the
mathematical theory, participants are highly encouraged to pursue other
related aspects - statistical, computational, applied, or otherwise -
of this exciting new field.
Knot Invariants and Categorification
Taught by
Morgan Opie and
Joshua Wang.
We'll begin with an introduction to knot theory via classical
knot invariants (genus, unknotting number, slice genus, etc.) with
many pictures and examples, including an introduction to the Jones
polynomial. The rest of the tutorial will focus on two generalizations of
the Jones polynomial, both having a categorical/algebraic flavor. The
first will be Khovanov homology, a theory which "categorifies" the
Jones polynomial in the same way that singular homology categorifies
Euler characteristic. Singular homology is a functor from the category
of topological spaces, and it recovers the Euler characteristic of a
(nice enough) space. In a similarly way, Khovanov homology will be a
functor from a category whose objects are knots, and it recovers the
Jones polynomial. The second generalization is to the knot polynomials
defined by Reshetikhin and Turaev using ribbon categories, and will be
covered at a pace and level of detail appropriate to the participants.
We will develop basic theory of categories as needed, focusing on the
visual theory of ribbon categories and its relationship to braids and
knots. We will make an effort to highlight aspects of category theory
that are relevant, but will not belabor formalities. This geometrically
motivated introduction will provide exposure useful for those interested
in future study of category theory. We do not expect students to have any
prior exposure to knot theory or category theory, but having familiarity
with basic algebraic topology (in particular, knowing the definition
and basic properties of singular homology) will be very helpful.
Elliptic Curves and Beyond
Taught by
Yujie Xu:
Elliptic curves are central objects in the study of number theory. There
have been many famous conjectures inspired by some explicit computations
involving elliptic curves, e.g. the Birch-Swinnerton-Dyer conjecture
relating the L-functions to the rank of rational points of elliptic
curves; Mordell-Weil theorem etc. Elliptic curves also played an important
role in the proof of many famous theorems, e.g. the proof of Fermat's
last theorem involves proving the modularity of some elliptic curves.
In this tutorial, we will first cover the foundations of elliptic curves,
and then, if time permits, we will cover some more advanced topics related
to elliptic curves such as modular curves and Galois representations,
Abelian varieties and their moduli spaces.